second form of Cauchy integral theorem


Let the complex function f be analytic in a simply connected open domain U of the complex planeMathworldPlanetmath, and let a and b be any two points of U.  Then the contour integral

γf(z)𝑑z (1)

is independent on the path γ which in U goes from a to b.

Example.  Let’s consider the integral (1) of the real part function defined by


with the path γ going from the point  O=(0, 0)  to the point  Q=(1, 1).  If γ is the line segmentMathworldPlanetmath OQ, we may use the substitution


and (1) equals


Secondly, we choose for γ the broken line OPQ where  P=(1, 0).  Now (1) is the sum


Thus, the integral (1) of the function depends on the path between the two points.  This is explained by the fact that the function f is not analytic — its real part x and imaginary part 0 do not satisfy the Cauchy-Riemann equationsMathworldPlanetmath.

Title second form of Cauchy integral theorem
Canonical name SecondFormOfCauchyIntegralTheorem
Date of creation 2013-03-22 15:19:39
Last modified on 2013-03-22 15:19:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 30E20
Synonym equivalent form of Cauchy integral theorem
Related topic CauchyIntegralTheorem
Defines example of non-analytic function